The World Cup winner has parted company with the Rossoneri after 13 years at San Siro, and is set to be offered the opportunity to continue his career in Argentina
The former Italy international is a free agent after lining out for the Rossoneri for the final time in Sunday's Serie A win over Novara at San Siro.
The 34-year-old midfielder has already claimed that he would be open to a return to Rangers but Boca have now entered the fray.
"Gattuso is a player we really like and the doors to our squad are always open for one as prestigious as he," Budna told Sky Sport Italia.
Meanwhile, Budna confirmed that Catania are keen on signing Boca attacker Pablo Mouche but claimed that the Sicilians face stiff competition for the 24-year-old's signature.
"Pablo is a great player and there have been talks with Catania, which have been going on for about a month," he revealed. "But there are other clubs interested in him, so there is nothing definite at the moment."
The aforementioned Gattuso joined Milan from Salernitana in 1999 but has seen little game time after recovering from career-threatening ocular problems over the past nine months and therefore decided against extending his contract, which expires in June.
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